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## Main Question or Discussion Point

H is a contravariant transformation matrix, M is a covariant transformation matrix, G is the metric tensor and G

I have G = 1/sin

I compute H = G*M where M = {(1 0), (cosα sinα)} and get H = {(1 -1/tanα),(0 1/sina)} which is what I expect.

Now I want go from H back to M so I compute M = G

So by my reckoning G

But when I multiply G

What am I missing?

^{-1}is its inverse. Consider an oblique coordinates system with angle between the axes = αI have G = 1/sin

^{2}α{(1 -cosα),(-cosα 1)} <- 2 x 2 matrixI compute H = G*M where M = {(1 0), (cosα sinα)} and get H = {(1 -1/tanα),(0 1/sina)} which is what I expect.

Now I want go from H back to M so I compute M = G

^{-1}HSo by my reckoning G

^{-1}= 1/sin^{4}α{(1 cosα),(cosα 1)}But when I multiply G

^{-1}and H I don't get back to M. The is a 1/sin^{4}α multiplying the whole thing.What am I missing?